PCUNGRQ(l)

📦 version 1.5, in ubuntu jammy

💖 PCUNGRQ 💖

okay bestie, sit down, hydrate, because we're talking about PCUNGRQ — that's the LAPACK routine from LAPACK version 1.5, and yes she is the one who builds you an orthonormal-row queen. This is her manual page, section l, and I'm gonna walk you through every single crumb of it because you deserve the full tea, not the summary. 🫶

✨ NAME ✨

PCUNGRQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N

Q = H(1)' H(2)'

like. she's giving reflector-girlboss energy. 💅

🦋 SYNOPSIS 🦋

SUBROUTINE PCUNGRQ( M, N, K, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

    INTEGER         IA, INFO, JA, K, LWORK, M, N

    INTEGER         DESCA( * )

    COMPLEX         A( * ), TAU( * ), WORK( * )

🌸 PURPOSE 🌸

PCUNGRQ generates an M-by-N complex distributed matrix Q denoting A(IA:IA+M-1,JA:JA+N-1) with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N

as returned by PCGERQF. (yes, PCGERQF is her upstream source, we love a collab)

📓 Notes 📓

Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location.

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

okay so here's the little legend of where everything lives, no cap:

  • DTYPE_A (global) — stored in DESCA( DTYPE_ ): the descriptor type. In this case, DTYPE_A = 1.
  • CTXT_A (global) — stored in DESCA( CTXT_ ): the BLACS context handle, indicating the BLACS process grid A is distributed over. The context itself is global, but the handle (the integer value) may vary.
  • M_A (global) — stored in DESCA( M_ ): the number of rows in the global array A.
  • N_A (global) — stored in DESCA( N_ ): the number of columns in the global array A.
  • MB_A (global) — stored in DESCA( MB_ ): the blocking factor used to distribute the rows of the array.
  • NB_A (global) — stored in DESCA( NB_ ): the blocking factor used to distribute the columns of the array.
  • RSRC_A (global) — stored in DESCA( RSRC_ ): the process row over which the first row of the array A is distributed.
  • CSRC_A (global) — stored in DESCA( CSRC_ ): the process column over which the first column of the array A is distributed.
  • LLD_A (local) — stored in DESCA( LLD_ ): the leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q.

LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.

Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

An upper bound for these quantities may be computed by:

LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

🎀 ARGUMENTS 🎀

  • M (global input) INTEGER — the number of rows to be operated on i.e the number of rows of the distributed submatrix Q. M >= 0.
  • N (global input) INTEGER — the number of columns to be operated on i.e the number of columns of the distributed submatrix Q. N >= M >= 0.
  • K (global input) INTEGER — the number of elementary reflectors whose product defines the matrix Q. M >= K >= 0.
  • A (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA+M-K <= i <= IA+M-1, as returned by PCGERQF in the K rows of its distributed matrix argument A(IA+M-K:IA+M-1,JA:*). On exit, this array contains the local pieces of the M-by-N distributed matrix Q. (glow up 💅 in and out, same array, different era)
  • IA (global input) INTEGER — the row index in the global array A indicating the first row of sub( A ).
  • JA (global input) INTEGER — the column index in the global array A indicating the first column of sub( A ).
  • DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
  • TAU (local input) COMPLEX, array, dimension LOCr(IA+M-1) — this array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCGERQF. TAU is tied to the distributed matrix A.
  • WORK (local workspace/local output) COMPLEX array, dimension (LWORK). On exit, WORK(1) returns the minimal and optimal LWORK.
  • LWORK (local or global input) INTEGER — the dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( MpA0 + NqA0 + MB_A ), where
IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW =
INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, MB_A,
MYROW, IAROW, NPROW ), NqA0 = NUMROC( N+ICOFFA, NB_A, MYCOL,
IACOL, NPCOL ),

INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. (she just peeks, she doesn't yell ✨)

  • INFO (global output) INTEGER
    • = 0: successful exit 🎉
    • < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i.
bestie, the negative INFO is not a vibe check failure on your part, it's just the routine telling you exactly which argument was acting up. read the number, fix the argument, try again. 💖

and that's the whole page, fr fr. she's PCUNGRQ, she's from LAPACK version 1.5, dated 12 May 1997, and she's been out here making orthonormal-row matrices since before you were born. iconic behavior honestly. 🎀🦋✨

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